\[\frac{d}{dx}(cosec^{-1}x)\]
let \[y=cosec^{-1}x\]
then \[cosecy=x\]
.Differentiating implicitly gives
\[-cosecycoty \frac{dy}{dx}=1 \rightarrow \frac{dy}{dx}=- \frac{1}{cosecy coty}\]
.To express
\[\frac{dy}{dx}\]
in terms of \[x\]
use \[cosecy=x,\; coty = \sqrt{cosec^2y-1} =\sqrt{x^2-1}\]
.Then
\[\frac{dy}{dx}=- \frac{1}{x \sqrt{x^2-1}}\]
.